Tiling tripartite graphs with 3-colorable graphs

Abstract

For any positive real number γ and any positive integer h, there is N0 such that the following holds. Let N ≥ N0 be such that N is divisible by h. If G is a tripartite graph with N vertices in each vertex class such that every vertex is adjacent to at least (2/3 + γ)N vertices in each of the other classes, then G can be tiled perfectly by copies of K h,h,h. This extends the work in [Discrete Math. 254 (2002), 289-308] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that the minimum-degree (2/3 + γ)N in our result cannot be replaced by 2N/3 + h - 2.